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Solving the problem of physicsproblem solvingEileen Scanlon aa I.E.T., The Open University , Milton Keynes MK7 6AA,EnglandPublished online: 09 Jul 2006.

To cite this article: Eileen Scanlon (1993) Solving the problem of physics problem solving,International Journal of Mathematical Education in Science and Technology, 24:3, 349-358,DOI: 10.1080/0020739930240303

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INT. J. MATH. EDUC. SCI. TECHNOL., 1993, VOL. 24, NO. 3, 349-358

Solving the problem of physics problem solving

by EILEEN SCANLON

I.E.T., The Open University, Milton Keynes MK7 6AA, England

(Received 27 January 1992)

My thesis reported a detailed study of physics problem solving behaviour. Init, physics problem solving among novices was investigated, and features ofproblem solving behaviour which affect their success are identified. Why dostudents of physics have so much difficulty with physics problem solving? Oneview is that their difficulties arise mainly from other sources, such as weakness inproblem solving. These views were explored through a series of studies on thebehaviour of novice physics problem solvers. Protocols of 259 solutions to avariety of kinematics and dynamics problems were collected. These protocols, thework of 35 subjects with varying amounts of familiarity and expertise in problemsolving were analysed. The protocols were examined for a number of featureswhich are hypothesized to play a role in a subject's success or failure in solving aparticular problem. The features examined included concept understanding aswell as other aspects of the problem-solving process such as (i) reading behaviour,(ii) attitude to solving problems, (iii) equation solving ability, (iv) graphconstruction and interpretation behaviour, and (v) physics knowledge of differenttypes from different types from different sources.

1. IntroductionDeveloping confident successful problem solving in students who are beginning

their study of basic physics is a problem. Such students get lots of simple physicsproblems wrong even though on the surface all they need to do is remember a fewequations, substitute appropriate values and perform some simple manipulations ofthe equations. Physics teachers, and indeed their pupils, would like to know what it isthat makes physics problem solving so difficult.

My own interest in this topic arises from 12 years teaching introductory schooland college physics. The particular event which sparked off this programme ofrresearch was my experience of a group of adult students enrolled on the OpenUniversity's science foundation course. After a 4-week, 40-hour intensive study ofbasic kinematics many of the students were unable to complete simple problems.What made these problems so difficult—was it the physics or was it the problemsolving?

Much has been written about what makes physics difficult to learn. The modernsubject of physics developed in the last century out of the much older subject ofnatural philosophy. Within that discipline the study of motion, kinematics, has along history, featuring work done by Aristotle, Galileo, Newton and Einstein. So,the physics concepts to which students are introduced have a history themselves.

Methods of instruction in physics have been influenced by psychological andother theories of learning. Piaget's stage theory has been used to construct a sequenceof concepts appropriate to the various stages of children's development he proposes.Discovery methods have been applied to physics teaching, in particular to the role oflaboratory experience in the formation of concepts [1]. Physics teachers have alsotried instruction in the history of the science they are trying to teach as the mostcompelling way of presenting new concepts. These last two methods (the discovery

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350 E. Scanlon

and the historical) share the assumption that it is important for students to learn whatphysics is as well as how to do it. So either by acting as scientists themselves (in thediscovery method) or learning about how scientists in the past developed the ideascurrently in use (the historical method), students will learn what the purpose ofscientific investigations are and come to share the set of skills which physicistspossess. Physicists talk about 'thinking like a physicist' but it is difficult to get themto be precise about what this phrase means.

Physicists deal with concrete situations but in doing so invent abstract conceptswhich can sound deceptively simple to students. In using these concepts, studentsfind themselves working with mathematical formulae and representations and sohave an additional stumbling block. .

The hypothesis that any subject can be taught effectively in some intellectuallyhonest form to any child at any stage of development [2] has provided the rationalefor curriculum reformers to introduce more abstract and difficult materials tochildren at younger ages. During the lifetime of several curriculum projects, both inBritain and the USA, researchers attempted to produce innovative teachingmaterial—Nuffield science in England, and the Physical Science study commissionand Harvard Project physics in the USA. Hegelson et al. [3], reporting on a NationalScience Foundation statistical survey, noted that a maximum of 40% of teachers,over the time period 1955—1975, had switched textbooks to use any project materials.In Britain relatively few schools offer Nuffield courses. In any case, a survey of howscience textbooks were used by sixth-formers discovered that only about one-thirdof the main physics text was read and that this use was mainly outside formal lessonsfor questions and exercises [4]. Interestingly, where Nuffield texts were used inchemistry teaching the average amount read shot up dramatically to over 80%. So,although there has been some curricular reform, it can be argued that much physicsteaching remains similar to a more traditional science teaching approach. Inparticular, there tends not to be specific instruction in 'problem solving'. Problemsolving skills are supposed to emerge naturally via the students' experience overmany physics problems.

The fact that this is a common view was borne out by an informal survey of somephysics teachers' views of physics problem solving carried out in the south ofEngland.1 An exploratory questionnaire was sent out to 25 secondary school physicsteachers in the counties of Bedford and Northampton, which provided someinteresting insights on the attitude an opinions that practising teachers have on theteaching of physics problem solving. Among other things the teachers were asked tocomment on the statement 'The greatest problem in physics teaching is notdeveloping concepts but in helping students to apply the knowledge they alreadyhave in problem solving situations'. One commented

It may be that the difficulty my students experienced was with the problemsolving rather than the physics concept they were using—that they had theknowledge but not the skill to choose what knowledge was appropriate to agiven situation, and what was required to find a solution to the problem and torecognise that solution when it appeared. This involves knowing the extent to

1 The empirical work described in this thesis was completed vefore the introduction ofG.C.S.E.

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Solving the problem of physics problem solving 351

which general problem solving skills exist and whether it is possible to teachthem explicitly or implicitly to students.

The respondents were divided about the importance or even possibility of teachingproblem solving.

2. The research questionsIn the thesis, the following question was addressed 'How can beginning students

of physics best be encouraged to develop confident problem solving strategiessufficient to solve simple kinematics problems? This question subdivides into severalothers:

(a) What kind of strategies do students employ in practice?(b) Are these strategies ineffective?(c) What are effective strategies?(d) Can students who employ ineffective strategies be taught to apply more

effective ones?

The thesis concentrated on questions (a) and (b).

3. The empirical workAt the time this work was started, the literature on physics problem solving told a

confusing story which was at odds with my own experience as a physics teacher. Themost sensible way forward was thus to undertake empirical work in which studentswere observed solving a variety of problems. Protocols (detailed records ofeverything written or said while working) of 259 solutions to a variety of kinematicsand dynamics problems were collected. These protocols, the work of 35 subjectswith varying amounts of familiarity and expertise in problem solving were analysed.The resultant protocols were analysed with a view to answering questions (a) and (b).Some aspects of the students' work, e.g. their use of graphs, proved to be especiallyinteresting. These issues were explored in much greater detail via the construction ofproduction rule models of the students' behaviour. This will not be further discussedhere. See [5] for more information.

The thesis thus contains a detailed empirically based account of physics problemsolving which could be used as a basis to address questions (a) and (b) above.

The thesis was a part time one. The first work with adults took place in March,1981 and empirical work and modelling continued till 1985.

A review of the multi-faceted literature on (i) previous work on conceptdevelopment in physics, (ii) the nature of problem solving in physics and how toteach it and (iii) instructional design based on cognitive models of physics problemsolvers shows that there is not a consensus about physics problem solving and thatmany previous studies lack a firm empirical base. The methodology of the workquoted in the thesis, in particular the protocol collection which forms the major partof the experimental work on which the results are based, derives from the Carnegie-Mellon school. The current status of protocol data in cognitive psychology as validdata and the technique of production rule modelling used to construct models of theobserved behaviour of problem solvers are part of this school. The experimental set-up used to collect combined paper and pencil records and think aloud verbal data isalso described in the complete thesis, as is protocol collection and production rulemodelling, and a description of the protocols of physics problem solvers on which the

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352 E. Scanlon

conclusions of this book are based, but in this summary article only the conclusionsof the work will be discussed fully. Protocols of problem solvers were obtained fromsubjects of a variety of ages, whose physics background ranged from introductoryphysics to graduate level study, working alone or in pairs to a variety of think aloudinstructions. From these protocols, those of first year 'A' level students working inpairs were selected as the most productive research environment. Students fromthree more schools were invited to work through these problems in pairs. The thesisexhaustively analyses students' performance on all the problems both on a problem-by-problem basis as well as a student-by-student basis. Also an overall description ofstudents' problem solving in answer to the questions given earlier, discussing theimplications of the protocols for a description of the problem solving process isgiven. The implications of the work for the design of physics instruction are alsodiscussed and so offers tentative answers to questions (c) and (d).

The literature review highlighted the importance of detailed observation of areasonable number of subjects engaged in problem solving. Table 1 gives anoverview of the data collection activities involving 35 subjects and pairs of subjectson an extensive problem set.

4. An example of the empirical dataHere is an example of the kind of data that was collected. One problem in the set

'A car travels 200 m in 5 s, what is its average velocity?'

was intended to be a warm-up question to help the students gain confidence. Most ofthe subjects did get this question right. It gave the opportunity of finding out whatthe students thought was meant by average velocity. Some students had connectedthe idea of 'average' having something to do with 'total' and some had not. Figure 1shows one pair of students who got into severe difficulties before eventuallystruggling their way through to a (correct) answer. They first of all tried to use one ofthe constant acceleration equations and then consider the notion of average as having

Subject pool Title of episode Focus of interest Number of subject:

Open University Pilot adult Does think aloud work? 7first year What strategies do adult

solvers adopt?

Mixed group Pilot methodology Refine methods 6Collect protocols under

a variety of methods

First year Pairs Detailed examination 15A level students of problem solving behaviour

Graduates Experts Is there any distinctively 7or near graduates expert behaviour exhibited?with physicsin their degree(includes twophysics ph.D.s)

Table 1. Summary of observational studies.

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Solving the problem of physics problem solving 353

200 m in 5 s, average velocity . . . v = u . . . oh.

v2 = u2-2ax.

No, we don't use that one. Average velocity—that is something over 2, isn't it?

Yes, no, hang on a second.

What's that equation where you've got something over 2.

Over 21

You know, average velocity, there is an equation for it.

There are two equations for velocity aren't there?

v = u + at, can't use that one

Why not?

Because you haven't got acceleration.

OK, write down what we have got -—we've got time, which is 5 s.

t=5s

There may be more to the question than we think, right, which means we might have to work outacceleration first or whatever OK? Distance = 200 m. Now if we were going to work outacceleration, initial velocity is, we don't know

s = 200mHold on, what is the equation where you've got something over 2?

Distance travelled over 2? ... No, what's the definition of velocity? Rate of change of displacement

Oh, can't we just do it—if it travels 200 m in 5 s, how far does it travel in 1 s?

Yeah, you could, I suppose so, but would that give you the average? OK, try that.

Average velocity =200/5 = 40ms"1

-is equal to 200 over 5 which equals 40.

Yeah, that's right because you've got distance travelled over time taken, do you recognise that!?Figure 1. Protocol of a pair of sixteen-year-olds.

to do with something over two. Rejecting the use of the constant accelerationequations since they do not know the acceleration, they began to try to calculate theacceleration, and gave up! They tried to remember an equation which has 'somethingover two' but could not. They recalled the definition of velocity as the rate of changeof displacement without finding any inspiration. Then they appealed to notions of

'if it travels 200 m in 5 s how far does it travel in 1 s.'

The extract from their protocol shows that this pair had strong notions about theconcepts and the problem solving process which guided their actions. One is thataverage means something over two. The main expectation is that problems areusually harder than they appear on the surface.

5. The nature of the problem solving processThe most suprising feature of the results is that the less-experienced students

were relatively quite successful in solving the problems. The difference in

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354 E. Scanlon

performance between the experts and the novices was that the experts were moreconsistent in solving the simpler problems rather than that they were strikingly moresuccessful in solving the more difficult problems. The features of the problemsolving process were examined in terms of a framework within which students'behavior at each stage of the problem solving process was discussed. These were:

(a) Difficulties in knowing how to go about finding a solution. Behaviour to do withreading the problem statement and the subjects' general attitude to doingthe problems.

(b) Difficulties in knowing what to do once you've started. These problems includeunderstanding the subject matter and problems with applying an appropri-ate strategy.

(c) Difficulties in finishing appropriately. When and how to check and how toselect the appropriate units.

5.1. How to go about itStudents read and re-read the problems to get clues about what to do. Subjects

re-read the whole problem, sentences and parts of sentences. A framework of typesof reading event was developed. A 'reading event' means a clear signal in the verbalprotocol that the subject had interacted with the problem statement. Most of thereading events (approximately 30%) were a complete read through of the wholeproblem. The most noticeable difference beteen the novice subject pairs and theexpert singles was the percentage of events where the question to be answered wasread, 50% of the expert singles events compared to 30% of the novices. These figuressuggest that one key problem for novices is to remain sufficiently aware of the endresult which they are trying to achieve. The differences quoted in the literature aboutthe direction of working observed in novice/expert studies seems relevant to thisdiscussion. It has been claimed that novices work backwards [6] but I found thatnovices do not look back at the problem statement to check what quantity they aresupposed to be finding. It seems unlikely therefore that they would remember thequantity to be found from the first reading.

Another class of behaviour was apparent at this initial stage. This was behaviourinfluenced by a whole host of expectations which existed for the subjects about whatphysics problems were like and what doing physics problems is like. Severalsurprising expectations were expressed by the children. These include:

(a) physics problems are harder than they appear in the surface;(b) physics problems are usually solved by circuitous routes;(c) physics problems ought not to require the student to remember equations;

and(d) physics problems ought not to require the combination of equations dealing

with different quantities.

The first of these is related to comments like 'It must be harder than this' which litterthe protocols. Since most of the subjects had some experience of solving problems,one possible interpretation is that these subjects were simply used to more difficultproblems. Another possibility is that these comments refer to the circuitous routethat students sometimes have to follow to solve a problem, e.g.'we might have to findthe acceleration first or something like that'.

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Solving the problem of physics problem solving 355

Many subjects seemed appalled that they were being expected to remember anyequations. Some of the more experienced adults coped with their limited store ofremembered equations by working out problems from first principles, e.g. workingfrom definitions of velocity and acceleration rather than from remembering constantacceleration equations. The problem set as constituted presented some problems. Agreat deal of surprise was generated by problems where subjects had to combineequations about force with equations about velocity and acceleration. Theseproblems are not uncommon in physics textbooks but were regarded by the novicesas completely against the informal rules about what physics problems are like. One ofthese is that kinematics and mechanics equations ought not to be mixed up. Thismakes sense if each problem type cues its own specialized problem solving processes.

There were some interference effects from preceding problems in the set. Forexample, a problem where g was used to represent grams preceding a problem whereg was used as the acceleration due to gravity caused occasional confusion. On oneparticular problem, the repeated generation of v2 — u2 = 2ax, an inappropriateequation, may have been due to the fact that it was the only constant accelerationequation not used in the problem set by that point.

5.2. What you do once you get startedOnce the subjects had started the problems two sorts of difficulty arose—one set

arising from the physics content of the problems, and one from the strategies used toattempt them.

Difficulties with the content in physics problem have usually been discussed inthe literature in terms of student's misconcepts. This term has been used to meanboth wrong views and alternative views of how these concepts work. Researchersdiffer on the extent to which these alternative views are coherent, internallyconsistent [7] or fragmentary [8]. In the present study, evidence of a sloppiness ofexpression was found, supporting [8]. Subjects haphazardly interchanged descrip-tions of quantities such as metres per second and metres per second squared—bothnovices and experts. This did not seem in itself to be symptomatic of a similarconfusion between distance, velocity and acceleration. However, there was littleappreciation of the distinction between average velocity and instantaneous velocity.A sentence like 'A car starts from rest and after 3 s is moving with a speed of 60 m s~1

after one second' is often translated as 'A car is moving at 60 m s~1>, even by subjectswho at other times show no sign of confusing average and instantaneous velocity.There was no evidence that many subjects understood what it actually meant for anobject to be accelerating, though they could actually handle equations with the lettera in them competently and calculate a numerical value for acceleration. Describing abody as undergoing uniform acceleration seems unnecessary since they never docalculations which would involve non-uniform accelerations. Almost none of thesubjects ever refer to equations of motion as constant acceleration equations. (To beprecise, only two of the experts from the whole population of 35.)

5.3. StrategiesThe main conclusion of the pilot study carried out with Open University adults

was that they showed evidence of possessing physics problem solving strategies butno evidence of knowing which ones were appropriate or might be productive. Ifound examples of adults who 'always draw a diagram', 'always write down

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356 E. Scanlon

everything I know', 'always write down sentences about the problems'. Some similarexamples of strategies are used by the more experienced subjects. However, amongthe pairs of school children studied no similar strategies for dealing with theproblems were exhibited. Examples of writing down either the knowns or theunknowns was uncommon and no sign of having had any previous advice on how toproceed on these problems was apparent with one exception.

The one strategy which is in common use is that of drawing a graph—often theschoolchildren drew graphs when they couldn't think of what else to do. Graphswould often have the meaning of their axes shifted in full flight, e.g. a subject woulddraw a distance-time graph and then treat it as a velocity-time graph and vice versaor would draw a distance—time set of axes but fill in details of velocity versus time.Having drawn the graph, the subject would often not know what to do with it. Thetype of knowledge which is securely known by all the subjects is extremely over-generalized, e.g.:

(a) you can read something off from a graph using lines drawn across,(b) gradient equal 'something', and(c) 'something' is the area under the graph.

Such over-generalized rules allow enormous scope for continuing to work on aproblem. There was no sign of subjects being aware of any connection betweengraphs and the equations of motion they represent. Indeed, when one of the moreexperienced subjects worked out the relationship between the equation s = ut+%at2

and his distance-time graph, he was very surprised and pleased with himself. Oneproblem which required the average velocity to be read off from a velocity-timegraph, was completed in this way by only a small proportion of the subjects. In fact, aclose examination of this particular question suggests that the teaching of multiplerepresentations, when neither is adequately understood could be a hindrance ratherthan a help. The production rule modelling associated with the graph task providesan account of why there should be so much 'overhead' in switching representations.

Other examples of strange features noticeable in the solution paths of the subjectwere:

(a) a tendency to work some things out before writing them down;(b) a tendency to keep writing down decimal places ad infinitum; and(c) a tendency to play games with numbers until an answer is reached.

5.4. FinishingNone of the inexperienced subjects ever estimated what the result should be.

Only in very few protocols was any checking done but none of the subjects made anyattempt at principled checking, e.g. one suitable strategy would be double checkyour results by working it out by another route. The answer to problem C was 7^.This was rejected by two subjects who decided to check their arithmetic-the non-neatness of the answer worried them. Only one subject used the method of checkinghis units in order to check his answer and none used dimensional checking. Therewas an overall impression of engaging in a procedure which would gain marks, ratherthan seriously questioning whether an answer was right or wrong. This relates to anoverall impression throughout the protocols of writing things down to gain marksrather than using the paper to reflect one's thinking about the topic. This funding issimilar to comments in [9]. In some protocols very little of what is said is reflected inwhat gets written down.

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Solving the problem of physics problem solving 357

This analysis of the problem solving process in terms of a number of stagessuggests another instructional consequence: reflection of these distinct stages by thestudent could improve student's performance.

5.5. Implications for teaching of this description of problem solvingThere are three conclusions which can be drawn from this empirical work. The

first is that it could be useful to teach some aspects of problem solving skill explicitly.For example, it could be useful to teach the stages of problem solving just described.It could be helpful to emphasize the importance of reading the problem sufficientlyor the need for including a checking phase to decide whether the answer makes sense.The difficulties experienced by students switching between media is another areawhich direct teaching might help. The second conclusion also deals with the issue ofusing multiple representations of a situation. If pupils need help in understanding aproblem situation, then the addition of an extra representation may not help and mayeven hinder their understanding. The third conclusion is that expert performancemay not be a good model to preach to novice physics problem solvers. In fact, theirperformance is too opportunistic to function as a good model for novices.

6. SummaryI have explored the process of physics problem solving. To do so I have

concentrated on observing subjects solve fairly simple, mostly quantitative problemsof the type set for homework and in examinations. Let us consider the putativeambitions of a physics or science teacher entering a classroom at 9.30 on a wet wintermorning. Collins and Schiapin [10] suggest goals might be:

(a) to enable children to pass examinations in school subjects;(b) to show future citizens the nature of science;(c) to begin to teach future scientists how to do science; and(d) to teach children about some features of the natural world.

These are worthy aims but whether the first of these is achievable with the next threedepend, among other things on the type of questions set. Solomon [11] writes

Though teaching about science in our schools is in its infancy it is beginning tokick. We shall know it has come of age when all the public examination papersin science include questions related to the living use of science alongside thosewhich test our pupils' ability to memorize facts and solve numerical problems.

So far we are not doing very well even at improving students' abilities at the latter.My work has been a small step in the progress towards understanding what makesphysics problem solving hard to do. If we do not understand why individual studentsfind this hard we cannot properly address the task of improving the quality of physicsinstruction.

References[1] DEWEY, J., 1907, Democracy and Education (New York: Macmillan Press).[2] BRUNER, J., 1960, The Process of Education (Cambridge, Massachusets: MIT Press).[3] HEGELSON, S. L., BLOSSER, P. E., and HOWE, R. W., 1977, The status of precollege

science, mathematics and social science education 1955-1975, Vol 1 Science Educ-ation, National Science Foundation report, Washington, DC.

[4] NEWTON, D. P., 1984, Br. J. Educ. Technol., 15 (1), 43-51.

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358 Solving the problem of physics problem solving

[5] SCANLON, E., and O'SHEA, T., 1987, Learning Issues in Intelligent Tutoring Systems,edited by H. Mandl and A. Lesgold (New York: Springer Verlag).

[6] LARKIN, J. H., 1980, The cognition of learning physics. Applied Psychology Report No1, Department of Psychology, Carnegie-Mellon University, Pittsburg.

[7] MCCLOSKEY, M., 1983, Mental Models, edited by D. Gentner and A. L. Stevens(Hillsdale, New Jersey: Laurence Earlbaum Assosciates), pp. 299-322.

[8] DISESSA, A. A., 1985, Proceedings of the 15th Annual Symposium of the Jean Piaget Societyon Constructivism in the Computer Age, Philadelphia.

[9] LAURILLARD, D., 1984, The Experience of Learning, edited by F. Marton, D. Hounsell.,and N. Entwistle (Scottish Academic Press).

[10] COLLINS, H., and SCHIAPIN, S., 1986, Science in Schools, edited by J. Brown (MiltonKeynes: Open University Press), pp. 71-77.

[11] Solomon, J., 1986, In Science in Schools, edited by J. Brown (Milton Keynes: OpenUniversity Press), pp. 141-147.

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